Reductive Shafarevich Conjecture

TL;DR

This paper proves the holomorphic convexity of certain coverings of complex projective varieties related to reductive representations, answering a question from 2012, and constructs the associated Shafarevich morphism with new methods.

Contribution

It provides a simplified proof of holomorphic convexity for coverings associated with reductive representations and constructs the Shafarevich morphism unconditionally with algebraic properties.

Findings

Proved holomorphic convexity of coverings for complex projective varieties.

Constructed the Shafarevich morphism for reductive representations.

Simplified proof avoiding reduction mod p techniques.

Abstract

In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $\rho:\pi_1(X)\to {\rm GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had previously proven this result in 2004 when $X$ is smooth. While our approach follows the general strategy employed in Eyssidieux's proof, it introduces several improvements and simplifications. Notably, it avoids the necessity of using the reduction mod $p$ method in Eyssidieux's original proof. Additionally, we construct the Shafarevich morphism for complex reductive representations of fundamental groups of complex quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.